Finite-Time Blow-Up Prevention by Logistic Source in Parabolic-Elliptic Chemotaxis Models with Singular Sensitivity in Any Dimensional Setting

In recent years, a lot of attention has been drawn to the question of whether logistic kinetics is sufficient to enforce the global existence of classical solutions or to prevent finite-time blow-up in various chemotaxis models. However, for several important chemotaxis models, only in the space two dimensional setting has it been shown that logistic kinetics is sufficient to enforce the global existence of classical solutions (see [K. Fujie, M. Winkler, and T. Yokota, Nonlinear Anal., 109 (2014), pp. 56--71], [J. I. Tello and W. Winkler, Comm. Partial Differential Equations, 32 (2007), pp. 849--877]). The current paper gives a confirmed answer to the above question for the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source in any space dimensional setting: $u_t=\Delta u-\chi\nabla\cdot (\frac{u}{v} \nabla v)+u(a(x,t)-b(x,t) u)$ and $0=\Delta v-\mu v+\nu u$ under the homogeneous Neumann boundary conditions in a bounded domain $\Omega \subset \mathbb{R}^n$, where $\chi$ is the singular chemotaxis sensitivity coefficient, $a(x,t)$ and $b(x,t)$ are positive smooth functions, and $\mu,\nu$ are positive constants. We prove that, for every given nonnegative initial data $0\not\equiv u_0\in C^0(\bar \Omega)$, the system has a unique globally defined classical solution $(u(t,x;u_0),v(t,x;u_0))$ with $u(0,x;u_0)=u_0(x)$, which shows that, in any space dimensional setting, logistic kinetics is sufficient to enforce the global existence of classical solutions and hence prevents the occurrence of finite-time blow-up even for arbitrarily large $\chi$. In addition, the solutions are shown to be uniformly bounded under the conditions $a_{\inf}>\frac{\mu \chi^2}{4}$ when $\chi \leq 2$ and $a_{\inf}>\mu (\chi-1)$ when $\chi>2$.